Pseudofinite-dimensional Schrödinger representations PRELIMINARY NOTES

M. Bays & B. Hart

August 24, 2016

Abstract

Following Zilber [Zil16], we obtain the Schrödinger representation of the (3-dimensional) Heisenberg algebra on the tempered distributions as an ultralimit of finite-dimensional representations of certain subgroups of the Heisenberg group. Moreover, we see that the corresponding Weil representation can be obtained this way.

0.1 Schwartz space and tempered distributions

Let S be the Schwartz space of complex-valued functions on ℝ with rapidly decreasing absolute value.

Recall that the topology on S is that given by the seminorms

p_{n,m}(ϕ) = sup_{x} | x^{n}δ^{m}ϕ(x) | .

Recall also that the subspace D ⊆ S of compactly supported smooth functions is dense in S.

Let S′ be the space of tempered distributions, the continuous dual space of S. Write the corresponding sesquilinear pairing as ⟨T, ϕ⟩ := T (\overline{ϕ}) for T ∈ S′ and ϕ ∈ S. We have the seminorms

∥T ∥_{ϕ} := | ⟨T, ϕ⟩ |

for ϕ ∈ S. A subset B of S′ is \underline{bounded} iff each ∥·∥_{ϕ} is bounded on B. We consider bounded subsets with the topology induced from the ∥ · ∥_{ϕ}. In fact this topology coincides with that induced from the strong topology on S′, but we will not explicitly use or define the strong topology. Closed bounded subsets of S′ are compact [Sch66, Chap.III,Thm.XII].

Recall [Rudin 7.12(d)] that every measurable function f : ℝ → ℂ of polynomial growth, i.e. bounded in absolute value by some polynomial, induces a tempered distribution

⟨f, ϕ⟩ := ∫_{ℝ} f(x)\overline{ϕ}(x)dx.

In particular this induces embeddings of S and D into S′, which are dense by [Sch66, Chap.III Thm.XV].

The space of distributions D′ is the continuous dual of D. We have topological embeddings

D ⊆ S ⊆ S′ ⊆ D′.

Recall that a bilinear map u : X × Y → Z of topological vector spaces is hypocontinuous if each u(x, ·) and u(·, y) are continuous, and {u(x, ·) | x ∈ B} is equicontinuous for any bounded B ⊆ X, so in particular the restriction of the pairing to B × Y is continuous, and similarly for B′ ⊆ Y bounded.

By [Sch66, Chap.III,Thm.XI;Chap.VII,Sect.4], the pairings D × D′ → ℂ and S × S′ → ℂ are hypocontinuous.

Let E be the space of infinitely differentiable functions ℝ → ℂ with the topology of compact uniform convergence of all derivatives. By [Sch66, Chap.V,Thm.III], multiplication of functions and distributions is hypocontinuous as a function E × D′ → D′.

0.2 Metric logic structure on S′

We view S′ as the union of its closed bounded subsets (forming a “bornology”). S′ is not covered by any countable set of bounded sets, nor is S′ the union of a chain of bounded sets.

Each closed bounded B ⊆ S′ is compact, and moreover is a complete metric space. In fact, a single metric suffices to metrise any B. Indeed, S is separable, so let (ψ_{i}^{0})_{i<ω} be a countable topological basis. Concretely, we can take ψ_{i}^{0}(x) := x^{i}e^{−x^{2}}. Then define on S′ the metric

d(T, T′) := ∑_{i} 2^{−i} min(1, | ⟨T − T′, ψ_{i}^{0}⟩ |).

This does not metrise any relevant topology on S′ itself, but it does metrise the topology on any bounded subset: Lemma 0.1. Let B ⊆ S′ be closed and bounded. Then the restriction of d to B is a complete metric on B, metrising the topology of B.

Proof. The topology on B is induced by the pseudonorms ∥ · ∥_{ϕ} and is complete with respect to them, so it suffices to see that for any ϕ ∈ S and any ϵ, there exists δ such that for T, T′ ∈ B,

d(T, T′) < δ ⇒ | ⟨T − T′, ϕ⟩ | < ϵ.

But indeed, the pairing is hypocontinuous, so ⟨T, .⟩ is equicontinuous on B − B; hence for a good enough approximation to ϕ as a finite linear combination ϕ_{0} of the ψ_{i}^{0}, | ⟨T − T′, ϕ − ϕ_{0}⟩ | < ϵ/2 for all T, T′ ∈ B. Then take δ such that

d(T, T′) < δ ⇒ | ⟨T − T′, ϕ_{0}⟩ | < ϵ/2.
□

Compact metric spaces are the analogue in continuous logic of finite structures in discrete logic, and most notions from continuous logic degenerate in such spaces. In particular, the ultraproduct of subspaces may be identified with the topological ultralimit, as we now explain.

Consider a set X and a set B of subsets which covers X and is closed under finite unions, and suppose d is a metric on X such that equipping each B ∈ B with the restriction of d makes it a compact metric space. The example we have in mind is X = S′, with B the set of its closed bounded subsets, and d defined as above.

Call Y ⊆ X \underline{bounded} if Y ⊆ B for some B ∈ B. For Y_{i} ⊆ B ∈ B and an ultrafilter U, recall lim_{i→U} Y_{i} = {lim_{i→U} y_{i} | y_{i} ∈ Y_{i}} ⊆ B. Now for Y_{i} ⊆ X, define lim_{i→U}^{B} Y_{i} = ⋃_{B∈B} lim_{i→U}(Y_{i} ∩ B).

For X = S′, this notion of “bounded ultralimit” does not agree with the usual topological ultralimit for even the strong topology on X, since unbounded ultrafilters in X can converge; for our purposes such convergence is pathological, so we restrict to bounded sets.

Now suppose Y_{i} ⊆ X are subspaces such that lim_{i→U}^{B} Y_{i} = X. If the Y_{i} are equipped with interpretations of a relation R^{Y_{i}} ⊆ Y_{i}^{n}, define the induced relation on X by R^{X} := lim_{i→U}^{B} R^{Y_{i}} ⊆ X^{n}. Then this notion of ultralimit of structures agrees with the notion of ultraproduct of metric structures, in the following sense. Consider (Y_{i}, R^{Y_{i}}) as a metric structure in the language with a sort for each B ∈ B, interpreted in Y_{i} as Y_{i} ∩B, and a real-valued metric predicate R′_{B} on each sort, interpreted in Y_{i} as the distance predicate R′_{B}(x) = d(x, R^{Y_{i}} ∩ B^{n}) = inf_{y∈Y_{i}∩B^{n}} d(x, y) (where we define d on X^{n} by d(x, y) = max_{i} d(x_{i}, y_{i}), say). Then the metric ultraproduct is isomorphic to (X, R^{X}) considered as a multisorted metric structure in the same way.

We apply this to (partial) operators X → X by considering their graphs, so we define F^{X}(lim_{i→U} y_{i}) := lim_{i→U} F^{Y_{i}}(y_{i}) whenever {y_{i}}_{i} is bounded, F^{Y_{i}}(y_{i}) are defined, and {F^{Y_{i}}(y_{i})}_{i} is bounded, if this definition yields a well-defined partial operator.

Say f^{Y_{i}} are \underline{provably bounded} if \{F^{Y_{i}}(y_{i})\}_{i} is bounded whenever \{y_{i}\}_{i} is bounded and the F^{Y_{i}}(y_{i}) are defined. Note that the composition of provably bounded operators is provably bounded.

Metric logic admits a Łos theorem, but the only consequence of this we will need is the following weak statement for exact equality of compositions of operators, which follows directly from the above definitions.

Lemma 0.2. Suppose F_{j}^{Y_{i}} are partial operators, and F_{1}^{Y_{i}} ∘ F_{2}^{Y_{i}} = F_{3}^{Y_{i}} holds U-often, and suppose that the ultralimits F_{1}^{X} and F_{2}^{X} are welldefined partial operators. Suppose also that the F_{2}^{Y_{i}} are provably bounded. Then F_{3}^{X} is a well-defined partial operator, and F_{1}^{X} ∘ F_{2}^{X} = F_{3}^{X}.

Proof. Restricting the index set, we may assume the equality holds for all i. Let {y_{i}}_{i} be bounded with y_{i} ∈ dom(F_{3}^{Y_{i}}). Then {F_{2}^{Y_{i}}(y_{i})}_{i} is bounded by provable boundedness, so lim_{i→U} F_{1}^{Y_{i}}(F_{2}^{Y_{i}}(y_{i})) = F_{1}^{X}(lim_{i→U} F_{2}^{Y_{i}}(y_{i})) = F_{1}^{X}(F_{2}^{X}(lim_{i→U} y_{i})). □

0.3 Schrödinger representations

The 3-dimensional real Heisenberg Lie algebra η is the central extension of the trivial Lie algebra on ℝ^{2} via the canonical symplectic structure on ℝ^{2}; it can be defined by saying it has basis (P, Q, E), with E being central and [Q, P] = E. The infinitesimal Schrödinger representation (with respect to Q) of η on the Schwartz space S ⊆ L^{2}(ℝ) is given by P ϕ(x) = −ϕ′(x), Qϕ(x) = 2πixϕ(x), Eϕ(x) = 2πiϕ(x). The dual representation on S′ is defined by ⟨XT, ϕ⟩ := ⟨T, −Xϕ⟩; note that this is consistent with the embedding S ⊆ S′, since η acts (formally) skew-adjointly.

The corresponding real Heisenberg Lie group N is the corresponding central extension of ℝ^{2}, and can be defined as triples (u, v, e) with group structure (u, v, e) ∗ (u′, v′, e′) = (u + u′, v + v′, e + e′ + (uv′ − u′v)/2). The corresponding exponential map is then exp(pP + qQ + tE) = (p, q, t). We have [exp(X), exp(Y)] = exp([X, Y]). Let χ : ℝ → ℂ be the character χ(x) := e^{2πix}. The corresponding Schrödinger unitary representation of N on S (and even on L^{2}(ℝ)) is given by

exp(pP)f(x) = f(x − p)
exp(qQ)f(x) = χ(qx)f(x)
exp(tE)f(x) = χ(t)f(x).

This can be considered as the exponential of the infinitesimal representation of η defined above; in particular, \frac{d}{dt} exp(tX)f(x)|_{t=0} = Xf(x) for X ∈ η and f ∈ S. The dual representation on S′ can be defined by ⟨XT, g⟩ := ⟨T, X^{−1}g⟩.

0.4 Cyclic Schrödinger representations

Let N_{m} be the finite subgroup of N generated by U_{m} := exp( \frac{Q}{m}) and V_{m} := exp( \frac{P}{m}). This group is presented by the relations [U_{m}, [U_{m}, V_{m}]] = 1 = [V_{m}, [U_{m}, V_{m}]] expressing that [U_{m}, V_{m}] is central.

Let S′_{m} := C_{m^{2}}^{∗} be the space of complex-valued functions on C_{m^{2}} := ℤ/m^{2}ℤ. We identify S′_{m} with the space of m^{2}-periodic functions on ℤ in the obvious way. Define a finite dimensional “cyclic” unitary representation of N_{m} on S′_{m} by

U_{m}f(k) := q^{k}f(k)
V_{m}f(k) := f(k − 1),

where q := χ( \frac{1}{m^{2}} ). So [U_{m}, V_{m}] acts centrally as multiplication by q.

We will show that the infinitesimal Schrödinger representation of η can be obtained as an ultralimit of these S′_{m}.

Embed S′_{m} into S′ as the subspace of m-periodic linear combinations of δ functions supported at \frac{1}{m} ℤ; namely, define

θ_{m}(f) := ∑_{k∈ℤ} f(k)δ_{\frac{k}{m}} .

This is easily verified to be an embedding of the representations of N_{m} on S′_{m} into the Schrödinger representation of N_{m} ≤ N on S′ defined in the previous section. Lemma 0.3. For any non-principal ultrafilter U, we have lim_{m→U}^{B} θ_{m}(S′_{m}) = S′.

Proof. Let T ∈ S′. By [Sch66, Chap.VI Sect.4], T is the limit of a sequence of smooth functions with compact support, say T = lim_{m→∞} g_{m}. Define m-periodic approximations h_{m} ∈ θ_{m}(S′_{m}) to g_{m} by sampling around 0, h_{m}( \frac{k}{m}) := \frac{1}{m} g_{m}( \frac{k}{m}) for ⌊ \frac{−m^{2}}{2} ⌋ ≤ k < ⌊ \frac{m^{2}}{2} ⌋. Then for any ϕ ∈ S, we have ⟨g_{m} − h_{m}, ϕ⟩ →_{m→∞} 0. So lim_{m→∞} h_{m} = T. Hence {h_{m}}_{m} is bounded, and the ultralimit along any non-principal ultrafilter is T. □

Let III_{m} := ∑_{l∈ℤ} δ_{ml} be the Dirac comb, consider the shifts by convolutions with δ functions δ_{\frac{k}{m}} ∗ III_{m} = ∑_{l∈ℤ} δ_{ml+ \frac{k}{m}}, and note that

θ_{m}(f) = ∑_{0≤k<m^{2}} f(k)(δ_{\frac{k}{m}} ∗ III_{m}).

Define the discrete Fourier transform on S′_{m} by

F(f)(p) = \frac{1}{m} ∑_{0≤k<m^{2}} f(k)χ( \frac{pk}{m^{2}} ).

Then the embeddings preserve F, i.e. F(θ_{m}(f)) = θ_{m}(Ff) where F on S′ is the Fourier transform defined by ⟨FT, Fϕ⟩ = ⟨T, ϕ⟩, where F on S is the usual Fourier transform defined by (Ff)(p) = ∫_{ℝ} f(x)χ(px)dx.

Indeed, F III_{1} = III_{1} by Poisson summation, and similarly F III_{m} = \frac{1}{m} III_{\frac{1}{m}}. Hence

Fθ_{m}(f) = ∑_{0≤k<m^{2}} f(k)F(δ_{\frac{k}{m}} ∗ III_{m})

= ∑_{0≤k<m^{2}} f(k) ((Fδ_{\frac{k}{m}} ) · (F III_{m}))

= ∑_{0≤k<m^{2}} f(k) ((χ ∘ ( \frac{k}{m} ·)) · ( \frac{1}{m} III_{\frac{1}{m}} ))

= \frac{1}{m} ∑_{0≤k<m^{2}} f(k) ((χ ∘ ( \frac{k}{m} ·)) · ( ∑_{0≤l<m^{2}}(δ_{\frac{l}{m}} ∗ III_{m})))

= ∑_{0≤l<m^{2}} \frac{1}{m} ∑_{0≤k<m^{2}} f(k) ((χ ∘ ( \frac{k}{m} ·)) · (δ_{\frac{l}{m}} ∗ III_{m}))

= ∑_{0≤l<m^{2}} \frac{1}{m} ∑_{0≤k<m^{2}} f(k) (χ( \frac{lk}{m^{2}} ) · (δ_{\frac{l}{m}} ∗ III_{m}))

= ∑_{0≤l<m^{2}} F(f)(l) · (δ_{\frac{l}{m}} ∗ III_{m})

= θ_{m}(Ff).

F is uniformly continuous on S′, since ∥FT ∥_{ϕ} = ∥T ∥_{F^{−1}ϕ}. So it follows that F commutes with the ultralimit.

Define on S′_{m} approximations to the derivatives of exp(qQ) and exp(pP),

Q := \frac{m}{2} (U_{m} − U_{m}^{−1})

P := \frac{m}{2} (V_{m} − V_{m}^{−1}).

Lemma 0.4. Q and P are provably bounded on the S′_{m}, and they induce operators on the ultralimit S′ defined by

(i) QT = (2πix) · T;

(ii) P T = − \frac{d}{dx} T.

Proof. (i) It suffices to show that if T = lim_{m→U} f_{m} with {f_{m}}_{m} bounded, then {Qf_{m}}_{m} is bounded and lim_{m→U} Qf_{m} = (2πix) · T, i.e. if ϕ = ϕ(x) ∈ S, lim_{m→U} ⟨Qf_{m}, ϕ⟩ = lim_{m→U} ⟨f_{m}, −2πixϕ⟩ for all ϕ ∈ S. From the definition of Q on S′_{m}, we have ⟨Qf_{m}, ϕ⟩ = ⟨f_{m}, −mi sin(2πx/m)ϕ⟩. Now lim_{m→∞} −mi sin(2πx/m)ϕ = −2πixϕ in S, since lim_{m→∞} −mi sin(2πx/m) = −2πix in E and multiplication is hypocontinuous. So by hypocontinuity of the pairing, lim_{m→∞} (⟨Qf_{m}, ϕ⟩ − ⟨f_{m}, −2πixϕ⟩) = 0. This implies provable boundedness and that the ultralimit is as required.

(ii) One can check by direct calculation that on each S′_{m} we have V_{m} = F^{−1} ∘ U_{m} ∘ F, and hence by linearity P = (F^{−1} ∘ Q ∘ F). Meanwhile, on S′ we have \frac{d}{dx} = F^{−1} ∘ (−2πix·) ∘ F. F is provably bounded since ⟨Ff_{m}, ϕ⟩ = ⟨f_{m}, F^{−1}ϕ⟩, and Q is provably bounded by (i).

So we conclude by Lemma 0.2.

□

0.5 Weil representations

SL_{2}(ℝ) ≅ Sp(2, ℝ) acts by automorphisms on η and on N, acting trivially on the centre and via the canonical action on ℝ^{2}. We will write the action in exponential notation, X^{g}.

Via the Stone-von Neumann theorem, this induces a projective unitary representation on S, which turns out to yield a “metaplectic” unitary representation of the double cover Mp_{2}(ℝ) of SL_{2}(ℝ) on S, such that X^{g}(ϕ) = (g̃ ∘ X ∘ g^{−1})(ϕ) for g̃ ∈ Mp_{2}(ℝ), g the image in SL_{2}(ℝ), X ∈ η, and ϕ ∈ S. The dual representation on S′ is then defined by demanding formal unitarity, ⟨g̃T, g̃ϕ⟩ := ⟨T, ϕ⟩.

SL_{2}(ℝ) is generated by F := [\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}] and G_{b} := [\begin{matrix}1 & b \\ 0 & 1 \end{matrix}] for b ∈ ℝ,

so to define the metaplectic representation it suffices to define it on the lifts of these generators to Mp_{2}(ℝ). Following [Fol89, p.179], we have for ϕ ∈ S

F̃ϕ(x) = ±χ( \frac{1}{8} )Fϕ(x)
G̃_{b}ϕ(x) = ±χ(−b \frac{x^{2}}{2} ) · ϕ(x),

where the signs depend on the choice of lift.

TODO: could try to be more explicit? Need a nice way to understand the group structure of Mp_{2}(ℝ)…

We have seen that F, and hence the action of F̃ on S′, restricts to an operator on each S′_{m}. The action of G̃_{b} on S′ is by multiplication by ±χ(b \frac{x^{2}}{2} ). For b = \frac{c}{d} a rational number expressed in lowest terms, this restricts via θ_{m} to a partial operator G̃_{m,b} on S′_{m} if d divides m^{2}, with domain D_{m,b} consisting of functions supported on dℤ/m^{2}ℤ ⊆ ℤ/m^{2}ℤ; so θ_{m}(D_{m,d}) ⊆ θ_{m}(S′_{m}) consists of m-periodic linear combinations of δ functions supported on \frac{d}{m} ℤ.

Explicitly, G̃_{m,b}(f)(k) := ±χ(b \frac{k^{2}}{2m^{2}} ) · f(k).

Call an ultrafilter U on ℕ \underline{ultradivisible} if for all d ∈ ℕ, dℕ ∈ U.

Lemma 0.5. For U ultradivisible, lim_{m→U}^{B} θ_{m}(D_{m,b}) = S′.

Proof. If d|m,

θ_{m}(D_{m,d}) ⊇ θ_{m}(S′_{m/d}),

so the result follows from ultradivisibility and Lemma 0.3. □

So for b ∈ ℚ, we could define G̃_{b} to be interpreted on S′_{m} as G̃_{m,b}, and G̃_{b} would then commute with the ultralimit along an ultradivisible ultrafilter. But for b ∈ ℝ, we must simultaneously approximate b by rationals.

So for b ∈ ℝ, given m ∈ ℕ, write m = e^{2}f where e ∈ ℕ and f is a squarefree integer, and define b_{m} := \frac{⌊be⌋}{e}. So for U ultradivisible, lim_{m→U} b_{m} = b. Then interpret G̃_{b} on S′_{m} as G̃_{m,b_{m}}.

Lemma 0.6. Let b ∈ ℝ. Then G̃_{b} is provably bounded on the S′_{m}, and for U ultradivisible, lim_{m→U}^{B} preserves G̃_{b}.

1 INNER PRODUCT? 7

Proof. First we prove provable boundedness. So suppose f_{m} ∈ D_{m,b_{m}} are bounded, let ϕ ∈ S, and let ϵ > 0. Then by hypocontinuity of the pairing, for all but finitely many values of b_{m}, |⟨G̃_{b_{m}}f_{m}, ϕ⟩ − ⟨f_{m}, G̃_{−b}ϕ⟩| =

| ⟨f_{m}, G̃_{−b_{m}}ϕ − G̃_{−b}ϕ⟩ | = | ⟨f_{m}, ±χ((b − b_{m}) \frac{x^{2}}{2} )ϕ(x)⟩ | < ϵ. So since

⟨f_{m}, G̃_{−b}ϕ⟩ is bounded, also ⟨G̃_{b_{m}}f_{m}, ϕ⟩ is bounded.

Now we consider the ultralimit.

If e^{2}|m, we have D_{m,b_{m}} = S′_{e}. So as in Lemma 0.5, lim_{m→U}^{B} θ_{m}(D_{m,b_{m}}) = S′.

So let T ∈ S′; then we may write T = lim_{m→U} T_{m} with T_{m} = θ_{m}(f_{m}) ∈ θ_{m}(D_{m,b_{m}}) and bounded.

Now since lim_{m→U} b_{m} = b, lim_{m→U} χ(b_{m} \frac{x^{2}}{2} ) = χ(b \frac{x^{2}}{2} ) in E.

Then by hypocontinuity of multiplication,

lim_{m→U} θ_{m}(G̃_{b}f_{m}) = lim_{m→U} ±χ(b_{m} \frac{x^{2}}{2} ) · T_{m} = ±χ(b \frac{x^{2}}{2} ) · T = G̃_{b}T,

as required. □

We have already seen the analogous result for F, and hence for F̃. Now for h̃ ∈ Mp_{2}(ℝ), we can write h̃ as a product of instances of F̃ and G̃_{b}, and define its interpretation on S′_{m} as the corresponding composition. So by the provable boundedness and Lemma 0.2, lim_{m→U}^{B} preserves h̃. So we conclude:

Theorem 0.7. Equipping S′_{m} with the interpretations described above of P, Q, and h̃ ∈ Mp_{2}(ℝ), the bounded ultralimit lim_{m→U}^{B} S′_{m} along an ultradivisible ultrafilter U via the embeddings θ_{m} is S′ with the standard Schrödinger and Weil representations.

0.6 Further directions

1 Inner product?

Ill-defined in general (e.g. delta paired with heaviside), but sometimes well-defined. No one choice of normalisation will suffice if we want finite values (e.g. delta with delta vs. delta with χ(px)). Boris: restrict to lagrangian cross lagrangian, with appropriate normalisation. Worth considering generalisations of this?

Mention [HH], who seem to want to get a global inner product.

2 Trace

If the trace of the representation of h̃ ∈ Mp_{2}(ℝ) is defined as in mathematical physics (namely what??), can we confirm that the trace is preserved by the ultraproduct, generalising last section of [Z]?

3 Feynmann integrals

Can we? Should we?

REFERENCES 8

References

[Fol89] Gerald B. Folland. Harmonic analysis in phase space, volume 122 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1989.

[Sch66] Laurent Schwartz. Théorie des distributions. Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. IX-X. Nouvelle édition, entiérement corrigée, refondue et augmentée. Hermann, Paris, 1966.

[Zil16] Boris Zilber. The semantics of the canonical commutation relation. 2016.